Saving animation with keyframe inserting location and rotation
You can insert the new animation to the keyframe of the armature who is changed. You can actually keyframe a new rotation of just one bone of an armature, so if you modified a bone rotation in pose mode you can then press I to insert keyframe and then choose rotation, and this will insert a new keyframe for the rotation of this bone.

I attached a .blend file where two armatures are there and adjusted the bone rotation of specific bones.